论文标题
通过使用Chebyshev多项式，证明了某些苯甲酸的Kekulé编号的一些猜想
Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomials
论文作者
论文摘要
在化学方面，Cyvin-Gutman列举了Kekulé数字的某些苯甲醇，并将其记录为OEIS上的$ A050446 $。该数字正是由递归定义的两个变量数组$ t（n，m）$ $ t（n，m）= t（n，m-1） + \ sum^{\ lfloor \ frac {n-1} {2} {2} \ rfloor} _ {_ {k = 0} t（k = 0} $ t（n，0）= t（0，m）= 1 $，用于所有非负整数$ m，n $。有趣的是，这个数字也出现在加权图，图形多型，魔术标记和单位原始矩阵的背景下，由不同的作者研究。在OEI上做出了一些有趣的猜想。这些猜想与$ t（n，m）$的行和列生成函数均相关。在本文中，给出了列生成函数的明确公式，这也是Bóna，JU和Yoshida研究的生成函数$ f（n，x）$。我们还通过使用第二类的Chebyshev多项式获得TRIG函数表示。这使我们能够证明所有这些猜想。
In chemistry, Cyvin-Gutman enumerates Kekulé numbers for certain benzenoids and record it as $A050446$ on OEIS. This number is exactly the two variable array $T(n,m)$ defined by the recursion $T(n, m) = T(n, m-1) + \sum^{\lfloor\frac{n-1}{2}\rfloor}_{k=0} T(2k, m-1)T(n-1-2k, m)$, where $T(n,0)=T(0,m)=1$ for all nonnegative integers $m,n$. Interestingly, this number also appeared in the context of weighted graphs, graph polytopes, magic labellings, and unit primitive matrices, studied by different authors. Several interesting conjectures were made on the OEIS. These conjectures are related to both the row and column generating function of $T(n,m)$. In this paper, give explicit formula of the column generating function, which is also the generating function $F(n,x)$ studied by Bóna, Ju, and Yoshida. We also get trig function representations by using Chebyshev polynomials of the second kind. This allows us to prove all these conjectures.
